Some recent progress on the stationary measure for the open KPZ equation

TL;DR

This paper reviews recent progress on understanding the stationary measure of the open KPZ equation, connecting historical and recent results in interface growth models with boundary interactions.

Contribution

It synthesizes various developments and insights into the stationary measure for the open KPZ equation, highlighting connections to foundational work and recent advances.

Findings

Review of joint work with A. Knizel and H. Shen

Connections to classical models like ASEP and boundary conditions

Summary of recent mathematical progress in the field

Abstract

This note, dedicated in Harold Widom's memory, is an expanded version of a lecture I gave in fall 2021 at the MSRI program "Universality and Integrability in Random Matrices and Interacting Particle Systems". I will focus on the behavior of the stationary measure for the open KPZ equation, a paradigmatic model for interface growth in contact with boundaries. Much of this will review elements of my joint work with A. Knizel as well as with H. Shen, as well as subsequent works of W. Bryc, A. Kuznetsov, Y. Wang, and J. Wesolowski and of G. Barraquand and P. Le Doussal. The basis for this advance is fundamental work of B. Derrida, M. Evans, V. Hakim and V. Pasquier from 1993, of T. Sasamoto, M. Uchiyama and M. Wadati from 2003, and of W. Bryc and J. Wesolowski from 2010 and 2017. I will try to explain how all of this fits together, without laboring details for the sake of exposition. Though this work does not directly follow from Harold Widom's own work, it (and a great deal of my research) is very much inspired by his and Craig Tracy's work on ASEP.